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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.bad_roots"></a><a class="link" href="bad_roots.html" title="Examples Where Root Finding Goes Wrong">Examples Where Root Finding Goes
    Wrong</a>
</h2></div></div></div>
<p>
      There are many reasons why root root finding can fail, here are just a few
      of the more common examples:
    </p>
<h4>
<a name="math_toolkit.bad_roots.h0"></a>
      <span class="phrase"><a name="math_toolkit.bad_roots.local_minima"></a></span><a class="link" href="bad_roots.html#math_toolkit.bad_roots.local_minima">Local
      Minima</a>
    </h4>
<p>
      If you start in the wrong place, such as z<sub>0</sub> here:
    </p>
<p>
      <span class="inlinemediaobject"><object type="image/svg+xml" data="../../roots/bad_root_1.svg" width="372" height="262"></object></span>
    </p>
<p>
      Then almost any root-finding algorithm will descend into a local minima rather
      than find the root.
    </p>
<h4>
<a name="math_toolkit.bad_roots.h1"></a>
      <span class="phrase"><a name="math_toolkit.bad_roots.flatlining"></a></span><a class="link" href="bad_roots.html#math_toolkit.bad_roots.flatlining">Flatlining</a>
    </h4>
<p>
      In this example, we're starting from a location (z<sub>0</sub>) where the first derivative
      is essentially zero:
    </p>
<p>
      <span class="inlinemediaobject"><object type="image/svg+xml" data="../../roots/bad_root_2.svg" width="372" height="262"></object></span>
    </p>
<p>
      In this situation the next iteration will shoot off to infinity (assuming we're
      using derivatives that is). Our code guards against this by insisting that
      the root is always bracketed, and then never stepping outside those bounds.
      In a case like this, no root finding algorithm can do better than bisecting
      until the root is found.
    </p>
<p>
      Note that there is no scale on the graph, we have seen examples of this situation
      occur in practice <span class="emphasis"><em>even when several decimal places of the initial
      guess z<sub>0</sub> are correct.</em></span>
    </p>
<p>
      This is really a special case of a more common situation where root finding
      with derivatives is <span class="emphasis"><em>divergent</em></span>. Consider starting at z<sub>0</sub> in
      this case:
    </p>
<p>
      <span class="inlinemediaobject"><object type="image/svg+xml" data="../../roots/bad_root_4.svg" width="372" height="262"></object></span>
    </p>
<p>
      An initial Newton step would take you further from the root than you started,
      as will all subsequent steps.
    </p>
<h4>
<a name="math_toolkit.bad_roots.h2"></a>
      <span class="phrase"><a name="math_toolkit.bad_roots.micro_stepping_non_convergence"></a></span><a class="link" href="bad_roots.html#math_toolkit.bad_roots.micro_stepping_non_convergence">Micro-stepping
      / Non-convergence</a>
    </h4>
<p>
      Consider starting at z<sub>0</sub> in this situation:
    </p>
<p>
      <span class="inlinemediaobject"><object type="image/svg+xml" data="../../roots/bad_root_3.svg" width="372" height="262"></object></span>
    </p>
<p>
      The first derivative is essentially infinite, and the second close to zero
      (and so offers no correction if we use it), as a result we take a very small
      first step. In the worst case situation, the first step is so small - perhaps
      even so small that subtracting from z<sub>0</sub> has no effect at the current working
      precision - that our algorithm will assume we are at the root already and terminate.
      Otherwise we will take lot's of very small steps which never converge on the
      root: our algorithms will protect against that by reverting to bisection.
    </p>
<p>
      An example of this situation would be trying to find the root of e<sup>-1/z<sup>2</sup></sup> - this
      function has a single root at <span class="emphasis"><em>z = 0</em></span>, but for <span class="emphasis"><em>z<sub>0</sub> &lt;
      0</em></span> neither Newton nor Halley steps will ever converge on the root,
      and for <span class="emphasis"><em>z<sub>0</sub> &gt; 0</em></span> the steps are actually divergent.
    </p>
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